W
orking papers
W
orking papers
José Alcalde and Antonio Romero-Medina
Fair School Placement
a
serie
d
WP-AD 2011-22Los documentos de trabajo del Ivie ofrecen un avance de los resultados de las investigaciones económicas en curso, con objeto de generar un proceso de discusión previo a su remisión a las revistas científicas. Al publicar este documento de trabajo, el Ivie no asume responsabilidad sobre su contenido.
Ivie working papers offer in advance the results of economic research under way in order to encourage a discussion process before sending them to scientific journals for their final publication. Ivie’s decision to publish this working paper does not imply any responsibility for its content.
La Serie AD es continuadora de la labor iniciada por el Departamento de Fundamentos de Análisis Económico de la Universidad de Alicante en su colección “A DISCUSIÓN” y difunde trabajos de marcado contenido teórico. Esta serie es coordinada por Carmen Herrero.
The AD series, coordinated by Carmen Herrero, is a continuation of the work initiated by the Department of Economic Analysis of the Universidad de Alicante in its collection “A DISCUSIÓN”, providing and distributing papers marked by their theoretical content.
Todos los documentos de trabajo están disponibles de forma gratuita en la web del Ivie http://www.ivie.es, así como las instrucciones para los autores que desean publicar en nuestras series.
Working papers can be downloaded free of charge from the Ivie website http://www.ivie.es, as well as the instructions for authors who are interested in publishing in our series.
Edita / Published by: Instituto Valenciano de Investigaciones Económicas, S.A. Depósito Legal / Legal Deposit no.: V-3640-2011
WP-AD 2011-22
Fair School Placement
*José Alcalde and Antonio Romero-Medina
∗∗Abstract
This paper introduces τ-fairness as a compromise solution reconciling Pareto efficiency and equity in School Choice Problems. We show that, by considering a weak notion of equity that we refer to as λ-equity, it is possible to contribute positively to solve an open debate, originated by the efficiency-equity trade-off of the schooling problem. We also suggest a slight modification to some allocative procedures recently introduced in the United States to compute τ-fair allocations and provide support for this (possible) reformulation.
Keywords: School Choice Problem, Fair Matching.
JEL codes: C78, D63, I28.
*
This paper supersedes our previous version titled “Re-Reforming the Bostonian System: A Novel Approach to the Schooling Problem,” (Alcalde and Romero-Medina, 2011b). The authors acknowledge Salvador Barberà, Estelle Cantillon, Aytek Erdil, Fuhito Kojima, Jordi Massó, Josep E. Peris, Pablo Revilla and José María Zarzuelo for their useful comments. We would also like to acknowledge the financial support of the Ivie and the Spanish Ministry of Education and Science under projects SEJ2007-62656/ECON (Alcalde) and ECON2008-027038 (Romero-Medina).
I. Introduction
In this paper, we propose a notion of fairness that reconciles the requirements of efficiency and equity in School Choice Problems. We call this notionτ-fairness. Additionally, we provide an allocation procedure that proposes, for each School Choice Problem, an allocation satisfying τ-fairness. This procedure allows any District School Board (DSB) to implement a τ-fair allocation according to the stated preferences.
Every year, many municipalities face the problem of allocating the available public school places among a large number of new applicants. Municipalities entrust this task to DSBs, which are concerned with how places are distributed through the systematic application of priorities and rules.
Some DSBs resort to School Choice Systems because of the difficulties asso-ciated with providing students with places in their schools of choice. A School Choice System first determines how to prioritize seemingly identical students. Then, it decides how priorities should be combined with the preferences of the students to generate an allocation. In this paper we assume that priorities are exogenously determined, and we concentrate on the allocation procedure.
There are two main allocation procedures used by the DSBs. The first proce-dure is known as the Boston mechanism (BM) because it was used in the Boston area until 2005. The BM is an example of a decentralized system for School Choice Problems. It provides efficient allocations based on student preferences. The second procedure, the Student Optimal Stable mechanism (SOSM) is the pri-mary example of a centralized system for School Choice Problems. The SOSM equitably balances the preferences of students with the priorities of the schools.
The choice of a specific procedure depends on the objective of the DSB. When efficiency is the main goal, the DSB might use the BM; in contrast, the SOSM is a useful option when the objective is to reconcile the preferences of the students with the priorities of the school in an equitable manner. Taking the priority lists as given, the classic notions of efficiency and equity might not be applicable to School Choice Problems.1 The classical concept of fairness in the School Choice
1Alcalde and Romero-Medina (2011a) explore how to prioritize students to avoid the equity-efficiency trade-off.
framework is based on the confluence of equity and efficiency. Balinski and Sön-mez (1999) proposed a reinterpretation of the stability concept used in matching markets as an equity criterion in the School Choice Problem. Following this ap-proach, we suggest a weaker notion of equity, which we call λ-equity. This new notion will allow us to defineτ-fairness as the conjunction of bothλ-equity and Pareto efficiency.
To define λ-equity, we follow an old tradition in (cooperative) game theory. Aumann and Maschler (1964) proposed an iterative process to argue that some instability should not be (credibly) taken into account. With this idea in mind, we proceed as follows. Any student can object to a given allocation by claiming that it fails to be equitable according to the priorities of the school. If a student objects to the initial allocation, she must propose an alternative allocation. If no student claims that the new allocation is not equitable, the initial proposal fails to meet the equity criterion. Otherwise, we dismiss the initial objection. If an allocation is objected to without a counter objection or is not objected to at all, it isλ-equitable. Note that the logic of the process is very similar to the definition of the Bargaining Set proposed by Zhou (1994).
The concept ofτ-fairness introduces two interesting new features to the School Choice Problem. First, each School Choice Problem has at least oneτ-fair alloca-tion. Second, there are systematic procedures for selecting aτ-fair allocation for each School Choice Problem. In this paper, we propose a simple mechanism that always provides aτ-fair allocation for each School Choice Problem. This proce-dure, which we call the Compensating Exchange Places Mechanism(CEPM), 2 operates in two phases and works as follows:
Phase 1. First, it proposes an equitable and, thus, λ-equitable tentative allocation. This can be accomplished by using the SOSM.
Phase 2. Because the above allocation might fail to be efficient, we consider a pure exchange economy where the agents are the students and the exchangeable 2Alcalde and Romero-Medina (2011b) introduce a family of matching procedures called the Exchanging Places Mechanisms, the elements of which propose a τ-fair allocation. In this pa-per, for simplicity of exposition, we select one specific rule, the CEPM, designed to (possibly) compensate students for thenegativeeffects induced by the “tie-breaking” lottery.
goods are the places. Finally, the initial endowment for each student is the place that was tentatively assigned to her in Phase 1.
Our contribution has to be understood in the context of the reforms established in the schooling systems. In fact, the game-theoretical analysis of the BM can be seen as the origin of important reforms in the admission systems used in elemen-tary schools in some areas of the US.3Simultaneously to the above reforms, other
changes have been suggested in different countries.4
This redesign effort has resulted in the adoption of the SOSM as the best op-tion for schooling systems (i.e., for almost one-sided markets with indifferences). However, as Abdulkadiro˘glu et al. (2009), Erdil and Ergin (2008), and Kesten (2010) agree, the welfare lost due to the adoption of the SOSM can be disturbingly large.
We have seen how the implementation ofτ-fair allocations requires minimal reforms in the SOSM and avoids its lack of efficiency while simultaneously re-specting a well-defined equity criterion. In fact, related ideas have inadvertently been proposed by some authors. These authors have either used a mechanism sim-ilar to the CEPM in order to compute the welfare lost due to the use of the SOSM (see Abdulkadiro˘glu et al., 2009) or proposed efficiency corrections of the SOSM (see, e.g., Kesten, 2010) such as the Efficiency Adjusted Deferred Acceptance mechanism (EADAM), which computes an allocation that we can now recognize asτ-fair.
In this paper, we propose to adopt the CEPM, a procedure for selecting aτ-fair allocation. The CEPM accomplishes this by introducing a pure-exchange market for places where students have property (or usufruct) rights to the places they have been assigned by the SOSM. The theory behind our proposal has already been used in many decentralized environments where individuals have endowments and there is room for improvement.
Although it is unusual to argue that students can exchange the places they have been assigned, there are similar situations in public administrations where agents 3For example, see the papers by Abdulkadiro˘glu et al. (2009) related to the New York City High School Match or Abdulkadiro˘glu et al. (2006) related to the Boston Public School Match.
4For example, see the papers by Biró (2008) on the Hungarian system, Feng (2005) on some high school processes in China, Shelim and Salem (2009) on the process in Egypt, or Ting (2007) for a study of the university access system in Taiwan.
are allowed to exchange their job positions. Therefore, in our opinion, there is legal support for the introduction of a centralized exchange of places to be phased into the allocation procedure.
Finally, it is worth noting that efficient mechanisms are being either imple-mented (as in the BM) or proposed (as in the EADAM proposed by Kesten, 2010) in the absence of the equity consideration that underlies the CEPM. In particu-lar, we know that the τ-fair allocations are not only λ-equitable but can also be characterized as the efficient allocations that all students weakly prefer to the ones determined by the SOSM.
We conclude this section by providing an overview of the rest of the paper. First, we introduce the basic model in Section II, and we formally state the general incompatibility between equity and efficiency in the traditional framework. Sec-tion III is devoted to introducing the main contribuSec-tion of this paper; we propose a formal definition of τ-fairness, and in Theorem 11, we also describe the applica-bility ofτ-fair allocations to any School Choice Problem. A formal (constructive) proof for Theorem 11 is provided in Section IV, which focuses on introducing the CEPM. The set ofτ-fair allocations is characterized in Section V. In Section VI, we consider the strategic properties of the CEPM. Finally, our main conclusions are gathered in Section VII. For simplicity, we relegate some technical proofs to the Appendix.
II. The School Choice Problem
This section is devoted to introducing several formalisms related to the School Choice Problem (SCP). We consider two sets of non-empty disjoint agents to be called students and schools. The set of students has n individuals, and is denoted
by S = {s1, . . . , si, . . . , sn}. The set of schools is denoted by C and has m
ele-ments (i.e.,C ={c1, . . . , cj, . . . , cm}).
Each school has a (fixed) number of places to be distributed among the stu-dents, which will be called its capacity. Letqcj ≥1denote the capacity of school
cj, and letQ = qc1, . . . , qcj, . . . , qcm
denote the vector summarizing the capaci-ties of the schools. Schools are also endowed a linear ordering prioritizing the set of students. Let πcj ∈ R
n be the students’ ordering for school c
the(m×n)-matrix summarizing these priorities. Formally,πcj is described as an
n-dimensional vector such that for eachk ∈ {1, . . . , n}, there is a unique student
sifor whomπcjsi =k; given this description, the j-th row for matrixΠcoincides
with vectorπcj.
Note that using our description, no school would consider a student to be in-admissible. Most DSBs impose such a restriction in the way that the schools rank their potential students. Nevertheless, our model could easily capture the possibility of a student being inadmissible: this can be accomplished simply by introducing a new variable for each school that has defined the priority level of the last admissible student.
In contrast, each student has linear preferences over the set of schools such that no student will consider two different schools as equivalent and, additionally, that no school is considered unacceptable by a student. Let ρsi denote the
rank-ing of the schools denotrank-ing the preferences of student si,5 and let Φdenote the
(n×m)-matrix summarizing these rankings. Note that our model assumes that each student considers all the schools to be admissible.6 Nevertheless, we can
also reformulate this model by assuming that each student might consider some schools to be unacceptable. The essence of this paper is the same in both frame-works.
Therefore, a SCP can be described by listing the elements above(S,C,Φ,Π, Q). We will say that an SCP is non-scarce whenever there are sufficient places to al-locate among all the students
X
cj∈C
qcj ≥n.
Given a SCP,(S,C,Φ,Π, Q), a solution for it is an applicationµthat matches students and places. Such a correspondence is called a matching. Formally,
Definition 1 Amatchingfor(S,C,Φ,Π, Q)is a correspondenceµ, applyingS∪C
into itself such that: 5i.e.,ρ
sicj = 3indicates that studentsiconsiderscjto be her third-best school.
6Here, we can also invoke the legislative regulations establishing that school attendance is compulsory for children of certain ages.
1. For eachsi inS, ifµ(si)6=si, thenµ(si)∈ C.
2. For eachcj inC,µ(cj)⊆ Sand|µ(cj)| ≤qcj.
7
3. For eachsi inS and anycj inC,µ(si) = cj if and only ifsi ∈µ(cj).
The central solution concept used throughout the literature is pair-wise sta-bility, introduced by Gale and Shapley (1962). In the present paper, we follow the suggestion of Balinski and Sönmez (1999) to identify this stability notion as a situation where the distribution of places among students is envy-free, which is why we say that a matching is equitablewhenever it captures this idea of envy-freeness. Under our considerations (i.e., each school is acceptable to any student and vice versa), equity is defined as follows.
Definition 2 A matching for(S,C,Φ,Π, Q), sayµ, is determined to beequitable
if there is no student-school pair(si, cj)such that
1. ρsicj < ρsiµ(si)
8and
2. |µ(cj)|< qcj orπcjsi < πcjsh for somesh ∈µ(cj).
To illustrate the function of Definition 2, let us consider a matching µ, and let us assume that student si prefers to study at schoolcj rather than her current
schoolµ(si). Ifsi has a priority higher than some of the actual students attending
schoolcj or if this school still has vacancies, then studentsi might claim that the
allocation process has beenunfair.
The concept of efficiency has also been analyzed in this framework. To intro-duce this concept appropriately, let us remember that the only role for schools is to provide educational services to students. Therefore, the natural notion of ef-ficiency, as proposed by Balinski and Sönmez (1999) for this framework, is the Pareto efficiency from the perspective of the students.
Definition 3 Given a School Allocation Problem,(S,C,Φ,Π, Q), matchingµis
Pareto efficientif for any other matchingµ0 6=µ, there is a student,si, such that
ρsiµ(si) < ρsiµ0(si).
7Throughout this paper,|T|will denote the cardinality of setT. 8Throughout this paper, we adopt the convention thatρ
Note that for any non-scarce SCP, stability and/or efficiency of a matchingµ
implies that for each studentsi,µ(si)∈ C.
A matching rule,M, is a regular procedure that assigns a matching to each SCP. Rule M is said to be equitable if for any given problem, it always selects an equitable allocation. Similarly, we say that a matching rule is Pareto efficient if its outcome is always Pareto efficient relative to its input. Clearly, there are stable matching rules. In fact, any of the versions of the deferred-acceptance al-gorithms proposed by Gale and Shapley (1962) assigns an equitable matching for the related SCP. In contrast, thenow-or-never ruleintroduced by Alcalde (1996) always selects a Pareto-efficient matching when the proposals are made by the students.
We first address the possibility of designing matching rules that always select
fair allocations(i.e., matchings that are equitable and Pareto efficient). As Propo-sition 4 states, reconciling the “fairness” notion with equity and Pareto efficiency might be an impossible task.
Proposition 4 There is no matching rule that selects an equitable and
Pareto-efficient allocation for each SCP.
Proposition 4 suggests the need for a new solution concept that accurately combines the notions of equity and Pareto efficiency.
III. τ-Fairness: A New Solution Concept
In this section, we propose a new solution concept for the SCP. This concept reduces the trade-off between equity and efficiency. The central idea is to restrict the statements of students that are considered “admissible” to induce inequity in an allocation.
It is important to precisely define the objections (made by a set of agents) that should be taken into account. This definition is at the essence of both the Bargain-ing Set introduced by Aumann and Maschler (1964) and the solution concepts that follow in this paper. The Bargaining Set is based on the idea that any agent who formulates an objection against an allocation should propose an alternative allo-cation fitting some properties. Then, if an agent objects to an alloallo-cation, any other
agent might formulate an objection against this new proposal in the same fash-ion. That is, any other agent mightcounter object. The λ-equity of an allocation requires that the following criteria are met:
1. No agent will object to the allocation (i.e., it is equitable), or 2. Any objection presented by an agent will be counter objected.
We capture the concept of (weak) equity by considering only objections against an allocation that cannot be counter objected to be valid. The following example illustrates this proposal.
Example 5 Let us consider the following SCP: S = {1,2,3}; C = {a, b, c};
Q= (1,1,1); and the ranking and priorities matrices are
Φ = 1 3 2 2 3 1 2 3 1 , andΠ = 3 2 1 2 1 3 1 3 2 .
Note that matchingµ, withµ(1) = a; µ(2) = b; andµ(3) = c, is not equitable because student 2claims that she has priority over student1 for schoola. Now, let us propose the following arrangement to student2:
“If you are able to propose a matching that you prefer to µ and if no other student would claim that the new proposal fails to be equi-table (as you did when µ was proposed), the new matching will be implemented.”
Student2cannot propose such a matching.
Therefore, the arguments of the Bargaining Set that are captured byλ-equity can be informally described as follows. Let us consider a matching µ. Then, any student is free to claim that this allocation fails to be equitable. Her objection must be supported by an alternative matching. The new proposal will be accepted only if no student is able to show, using identical arguments, that the new matching is also inequitable.
Definition 6 [Fair Objection]
Let(S,C,Φ,Π, Q)be a SCP, and letµbe a matching for such a problem. Afair objectionfrom studentsi ∈ Sagainstµis a pair(si, µ0)such that
1. ρsiµ0(si) < ρsiµ(si), and
2. |µ(µ0(si))|< qµ(µ0(s
i)), orπµ0(si)si < πµ0(si)sh for somesh ∈µ(µ
0(s i)).
Definition 7 [Counter Objection]
Let(si, µ0)be a fair objection to matchingµ. Acounter objectionfrom studentsh
against(si, µ0)is a pair(sh, µ00)that constitutes a fair objection to matchingµ0.
We say that (si, µ0) is a justified, fair objection to µ if it cannot be counter
objected.
Definition 8 [λ-Equity]
Let(S,C,Φ,Π, Q)be a SCP. We say that matchingµisλ-equitableif there is no
(si, µ0)constituting a justified, fair objection toµ.
Therefore, the idea ofλ-equity for matchingµis that whenever a student can claim that this matching is unfair, she must be unable to propose an alternative matching that no student would consider unfair.
Note that for any given SCP, the set ofλ-equitable matchings is a super-set of the sets of equitable allocations. Therefore, the following statement applies.
Proposition 9 Let(S,C,Φ,Π, Q)be a SCP. Then it has aλ-equitable matching.
In general, there are SCPs withλ-equitable matchings that are not equitable. Notice that the matching µ, proposed in Example 5, is not equitable, but it is
λ-equitable.
The central solution concept that we propose in this section,τ-fairness, com-bines two solution ideas, namely Pareto efficiency andλ-equity.
Definition 10 [τ-Fairness]
Let(S,C,Φ,Π, Q)be a SCP. We say that matchingµisτ-fairif it is Pareto effi-cient andλ-equitable.
The next question that we address is the existence ofτ-fair allocations. Al-though the sets of equitable and Pareto-efficient matchings might not intersect (Proposition 4), when we focus on λ-equitable allocations rather than equitable allocations, such an intersection is always non-empty.
Theorem 11 Let(S,C,Φ,Π, Q)be a School Allocation Problem. Then, it has a
matchingµthat isτ-fair.
IV. From Equity toτ-Fairness: The CEPM
This section introduces the CEPM, a matching rule that always selects a τ-fair allocation.
A simple way to define the CEPM is by the (sequential) composition of two well-known allocation procedures. The first, the input of which is a SCP, is the SOSM, which is the realization of the classic algorithm of students proposing deferred acceptance (see Gale and Shapley, 1962). When the SOSM is applied, we can interpret its outcome as a Housing Market (see Shapley and Scarf, 1974) whose agents are the students, and each individual is initially endowed with the place that the SOSM assigned to her. We can then apply Gale’s Tops Trading Cycle, introduced by Shapley and Scarf (1974), to reach a Pareto improvement related to the initial outcome of the SOSM. We show in Theorem 16 that the outcome for this iterative procedure is always aτ-fair matching.
Therefore, the CEPM can be seen as a constructive proof of Theorem 11, or, alternatively, as a suggestion for how to improve the system adopted by the Boston School Committee in 2005 by guaranteeing assignment efficiency.
We now provide a formal definition of the CEPM. First, Definition 12 de-scribes how to compute, for each SCP, its Student Optimal Stable matching,µSO.
Second, for a given problem, (S,C,Φ,Π, Q) and matching µ, Definition 13 de-scribes itsCompensating Placing Market. Then, for a givenplacing market, Def-inition 14 describes how the Tops Trading Cycleworks. We devote Appendix A to an example illustrating how to compute the CEPM for a specific SCP.
Definition 12 Let(S,C,Φ,Π, Q)be a SCP, we define itsStudent Optimal Stable
Step 1. Each student, si, applies to the school that is ranked first after ρi. Each
school, cj, tentatively accepts up to qcj students, according its priority list,
πcj. The remaining applications (if any) are rejected.
. . .
Step k. Each student,si, applies to the first school afterρi(if any) that has not
previ-ously rejected her application. If such a school does not exist,siwill remain
unassigned. Each school, cj, tentatively accepts up toqcj students,
accord-ing its priority list,πcj. The remaining applications (if any) are rejected.
The algorithm ends when each student who remains unassigned has been re-jected by all the schools. Each student is assigned to the school (if any) that accepted her application at the last step.
The Housing Market introduced by Shapley and Scarf (1974) involves a set of agents who each own one indivisible object (her house). Each agent exhibits pref-erences over the houses that can be described by a linear preorder. In this model, no agent considers two distinct houses as equivalent. Following this structure, we will build, from any SCP,(S,C,Φ,Π, Q), and matchingµ, a problem that shares the structure of the Housing Markets as described above. In this transition from a SCP and an allocation to a problem reflecting the structure of a Housing Market, there are some specifications that are (or can be seen as) natural.
In this placing market, a student, si, reveals that she wants to trade with sh
whenever her preferences for exchange satisfyshPsisi. Therefore, the following
can be uncontroversially assumed.
(a) Student si wants to exchange her place with sh only if she will garner a
positive benefit from such an exchange,
shPsisi =⇒ρsiµ(sh)< ρsiµ(si).
(b) For any two students,shandsk, who have been assigned to different schools,
place-ment in her preferred school,
ρsiµ(sh) < ρsiµ(sk)=⇒shPsisk.
Nevertheless, there is no a priori reason justifying a student’s prioritization of two different students who have been assigned a placement in the same school. A compensating placing market will consider that each student is willing to ex-change with an individual who is prioritized lower in that school. The rationale of such a hypothesis is derived from the manner in which priorities are established in real-life situations. When developing a priority list, schools divide students into four categories based on two main factors (siblings and residence). With this method of classifying students, multiple students are prioritized equally by the school at this stage. Lotteries are used to break these ties. Therefore, in most cases, it is expected that the school prioritizes one student relative to another be-cause of some random factor.9 The proposal of the Compensating Placing Market
to reverse the priority lists is intended to compensate for the random effect intro-duced by the lottery.10
Definition 13 Let (S,C,Φ,Π, Q) be a SCP andµ a matching for this problem.
We define its associated Compensating Placing Market, CP M(µ), as the pair
ˆ
S, PwhereSˆ, the set of agents, coincides with the set of students that, accord-ing toµ, has been assigned a placement in some school,
ˆ
S={si ∈ S :µ(si)∈ C}, and
P = (Psi)si∈Sˆ, thepreferences profile for exchange, satisfies the condition that
for eachsi ∈Sˆ,Psi is a linear preorder onSˆfulfilling the following: (a) For eachsh ∈Sˆsuch thatρsiµ(si) ≤ρsiµ(sh),siPsish;
(b) For eachsh, sk ∈Sˆsuch thatρsiµ(sh) < ρsiµ(sk),shPsisk; and
9Most systems consider several factors to categorize students. Then, a random lottery is used to break ties.
10Alcalde and Romero-Medina (2011b) explore a family of placing markets, including the CPM, each of which yields aτ-fair allocation. In the present version and for the sake of sim-plicity, we concentrate on the CPM.
(c) For any two studentssh andsksuch thatµ(sh) = µ(sk) 6=µ(si),shPsisk
if and only ifπµ(sh)sk < πµ(sh)sh.
In a more general context, aPlacing Marketis a pair (S, P), whereS is a set of agents andP denotes the profile of theirpreferences for exchange(i.e., for each
siinS,Psi is a linear preorder onS).
Definition 14 We define theTops Trading Cyclerule as the procedure for
assign-ing to eachPlacing Market,(S, P), the outcome for the following algorithm. Step 1. Let us consider the digraph whose set of nodes coincides withS, and for any
two (possibly equal) students,si andsh, there is an arc fromsi tosh ifsh
is the maximal forPsi inS. This digraph has at least one cycle. LetK(S)
be the set of students belonging to some cycle. Then, match each student inK(S)to her most preferred “mate for exchanging” (i.e., if si ∈ K(S),
thenT T Csi(S, P) = sh wheneversh is the maximal forPsiinS).
Let us defineS2 = S\K(S). IfS2 is empty, the algorithm stops. Other-wise, go to Step 2.
. . .
Step t. Let us consider the digraph whose set of nodes coincides with St, and for any two (possibly equal) students,siandsh, there is an arc fromsitoshifsh
is the maximal forPsiinS
t. This digraph has at least one cycle. LetK(St)
be the set of students belonging to some cycle. Then, match each student inK(St)to her most preferred “mate for exchanging” (i.e., ifs
i ∈K(St),
thenT T Csi(S, P) = sh wheneversh is the maximal forPsiinS
t).
Let us define St+1 = St\K(St). If St+1 is empty, the algorithm stops. Otherwise, go to Step(t+ 1).
Because there is a finite number of students, and for eacht, it holds thatSt+1 ( St, this algorithm stops in a finite number of steps.
We can now provide a formal definition for theCompensating Exchange Places Mechanism, which can be straightforwardly introduced by the next sequential pro-cedure
(1) Given the SCP (S,C,Φ,Π, Q), let us compute its Student Optimal Stable matching,µSO;
(2) Let us describe the Compensating Placing Market associated with the pair formed by(S,C,Φ,Π, Q)andµSO,CP M µSO
; and
(3) Let us compute the Tops Trading Cycle outcome forCP M µSO .
Once the process above is complete, we can distinguish two groups of students: first, those students who have not obtained a placement during the first phase (i.e.,
µSO(si) = si) and second, the remaining students. At the end of the process, the
former students will remain unassigned (i.e.,µCEP(si) = µSO(si) = si), whereas
the latter students will have the possibility of improving their initial assignment, namely µSO(s
i), if they are involved in the exchanging process guided by the
TTC rule.
Definition 15 [TheCompensating Exchange PlacesMechanism]
We define the Compensating Exchange Places Mechanism as the matching rule that assigns to each given SCP, say(S,C,Φ,Π, Q), the matching,µCEP, such that
for eachsi ∈ S, µCEP (si) = ( si ifµSO(si) = si µSO T T C si CP M µ SO otherwise . where
(a) µSO is the matching obtained when applying the SOSM to(S,C,Φ,Π, Q);
and
(b) T T C is the outcome for the Tops Trading Cycle rule when applied to the
Compensating Placing Market, CP M µSO
, assigned to the initial SCP,
(S,C,Φ,Π, Q), and its Student Optimal Stable matching,µSO.
We can now establish the following result: the CEPM can be applied to yield a τ-fair allocation. Therefore, Theorem 11 is a direct consequence of the result below.
Theorem 16 Let (S,C,Φ,Π, Q) be a SCP. Its Compensating Exchange Places matching,µCEP, is aτ-Fair allocation for such a problem.
A formal proof for this result can be found in the Appendix. Nevertheless, the reader might find a heuristic explanation for why our result holds useful.
First, let us note that theµSO outcome is (constrained) Pareto efficient when restricted to the set of equitable allocations; in other words, when considering a given problem,(S,C,Φ,Π, Q), for every student,si, and any equitable matching,
ρsiµSO(si) ≤ρsiµ(si).
Second, for each SCP,(S,C,Φ,Π, Q), and any matching,µ, the allocation,µ0, obtained by exchanging the places accordingly toT T C,
µ0(si) =
(
si ifµ(si) = si
µ(T T Csi(CP M(µ))) otherwise
.
is Pareto efficient and obeys the fact thatµdetermines school placement for stu-dents.
Together, the two observations above andµSO being equitable are crucial to the determination thatµCEP isτ-fair.
V. Characterizing the Set ofτ-Fair Allocations
In this section, we present a complete description of the τ-fair assignments of places.
The 2005 reforms in the Boston School System shifted the system from an efficient, decentralized process, the so-called BM, to an equitable, decentralized allocation mechanism, the SOSM.
If we concentrate on the trade-off between (BM) efficiency and (SOSM) eq-uity, we can select any (τ-fair) intermediate procedure minimizing this trade-off. As we pointed out in the previous section, this objective can be achieved by the CEPM. Nevertheless, as Theorem 17 reports, there are other ways to reduce the gap between efficiency and equity.
Theorem 17 Let(S,C,Φ,Π, Q)be a SCP andµbe a matching.µisτ-fair if and only if
(a) µis efficient, and
(b) For each student,si,
ρsiµ(si) ≤ρsiµSO(si).
Theorem 17 establishes that the only Pareto-efficient allocations that are τ -fair are those Pareto dominating the one that is implemented by the SOSM. Note that this result yields some straightforward consequences. The first consequence, which is the aim for Corollary 18, establishes that the unique allocation (if any) that might be equitable and Pareto efficient is the Student Optimal Stable match-ing. The second consequence, which is reflected in Corollary 19, reports that it is not worthwhile to distinguish between allocativefairnessandτ-fairnesswhen the former concept is non-empty. Therefore, the size of the set of τ-fair allocations can useful for measuring the welfare loss induced by employing the SOSM.
Corollary 18 Let(S,C,Φ,Π, Q) be a SCP, and letF(S,C,Φ,Π, Q) denote the
set of its fair matchings. Then
F(S,C,Φ,Π, Q)⊆
µSO .
Corollary 19 Let(S,C,Φ,Π, Q)be a SCP. If its Student Optimal Stable
match-ing is Pareto efficient, then anyτ-fair matching is a fair allocation.
VI. Strategizing the CEPM
We know that for every SCP, namely (S,C,Φ,Π, Q), its outcome (under BM) is Pareto efficient and gives the students incentives to act strategically. The school in which a student obtains a placement depends not only on her true characteristics but also on the preferences she has revealed. Additionally, for each SCP, the application of the BM combined with the students’ strategic behavior yields an (expected) equitable allocation that might fail to be Pareto efficient.
The use of the SOSM in the Boston area hinged largely on the realization that students will play the BM strategically, which will, at best, induce equitable allocations (see, e.g., Abdulkadiro˘glu et al., 2005, Section IV). Under the SOSM, students will reveal their preferences truthfully.
The CEPM is a combination of two strategy-proof mechanisms. First, we use the SOSM; then, we apply the TTC. However, the combination of both mecha-nisms cannot be strategy proof due to the impossibility of finding a Pareto-efficient and strategy-proof mechanism that Pareto dominates SOSM (see, e.g., Abdulka-diro˘glu et al., 2009 or Kesten, 2010).
The violation of strategy-proofness does not necessarily imply easy manipu-lability. However, it is not difficult to build examples of manipulation with the CEPM.
Example 20 Let us consider the following SCP. S = {1,2,3}; C = {a, b, c};
Q= (1,1,1); and the ranking and priorities matrices are
Φ = 1 2 3 2 1 3 3 1 2 andΠ = 3 2 1 1 3 2 2 1 3 .
The application of the student-proposed deferred acceptance algorithm, the output of which is µSO with µSO(1) = b, µSO(2) = a, µSO(3) = c.
There-fore, the CEPM yields matching µCEP with µCEP (1) = a, µCEP (2) = b, and
µCEP(3) =c.
Student 3 can misrepresent her ranking by declaring
ρ03 = (2,1,3).
If the CEPM is applied in such a case, this student is allocated a place at school
b. Therefore, in such a case, student 3 can manipulate the CEPM.
Let us observe that student 3’s manipulation is related to a (sophisticated) risky strategy. By declaring ρ03, she provokes that, at the SOSM phase, student
1 is assigned a placement at b. Simultaneously, she secures a placement at a, the best school from 1’s point of view but the worst school from her perspective.
Therefore, at the trading step, student 3 forces student 1 to prefer to trade with her, thus preventing student 2 from obtaining the unique placement available at
a. Because the SOSM is strategy-proof, when this student acts strategically, she never obtains a placement in a school that is better (from her point of view) than the one she would obtain by acting sincerely. Therefore, her only opportunity to gain from misrepresenting her preferences comes from the following combination of factors
(a) The student loses at the SOSM phase. However, due to this loss, she secures a placement at a school that is desired by other students.
(b) This desirability allows the student to gain during the trading phase.
Let us observe that the success of a student’s manipulation strongly depends on the information she has about the characteristics of her rivals.11 Roth and
Rothblum (1999) and Ehlers (2008) pointed out the difficulties associated with manipulation in incomplete information environments. In particular, the absence of complete information impedes the detection and exploitation of manipulation opportunities. For example, schools that are at a similar geographic distance might be almost identical from the student’s point of view. Additionally, the existence of opportunities to manipulate requires not only a consensus on the best alternative but preferences that are contrary to the general perception. If we assume that the preferences of the students are strongly positively correlated, this possibility decreases.
VII. Concluding Remarks
A philosophical position on the trade-off between equity and Pareto efficiency is at the origin of the modification to the mechanism used in the Boston area. The solution was to adopt the SOSM, which always selects an equitable allocation (see the description provided in Abdulkadiro˘glu et al., 2006). This reform precipitated a debate in school districts such as New York and San Francisco. At the origin 11In terms of the difficulty associated with manipulating the system, we want to stress that this strong requirement on knowing the characteristics of others is not necessary when the BM is employed.
of this debate was the objective to clarify, from a social perspective, the criteria behind the selection of a specific allocation rule.
The solution adopted in Boston did not solve the efficiency-equity trade-off that sparked the reform process. Moreover, there is evidence that the Boston re-form generated allocations that produced large welfare losses (see Abdulkadiro˘glu et al. (2009), Erdil and Ergin (2008), or Kesten (2010), among others).
This paper provides an alternative approach to circumventing the efficiency-equity dilemma. Our approach relies on a reinterpretation of the classical notion of stability in the framework of matching markets. We propose an alternative way to interpret the legitimacy of an agent’s claim against the equity of an allocation. The notion ofλ-equity embodies this idea and can be illustrated as follows:
When a student claims that an allocation fails to be equitable, she is asked to propose an “alternative allocation”. Then, any other student is asked whether, in her opinion, this alternative is equitable. A neg-ative answer would only be justified by proposing a third allocation, and so on. Therefore, the first claimant’s opposition to the DSB’s proposal might induce an everlasting chain ofobjectionsandcounter objections, or, eventually, this chain will end with the first claimant’s assignment of a place that she considers worse than that which she was initially assigned.
λ-equity requires that objections against the DSB’s proposal that can be counter objected should be disregarded. We find thatλ-equity is a weaker notion of equity and does not conflict with the objective of implementing a Pareto-efficient allo-cation. In fact, the combination of the ideas of λ-equity and efficiency is at the essence ofτ-fairness.
We have also found a simple way to obtain τ-fair allocations, the procedure called CEPM. This rule has a familiar flavor and can be implemented by introduc-ing a minimal reform in some schoolintroduc-ing systems, such as the procedure recently adopted in the Boston area. In short, we allow students to exchange the placements that were allocated to them in the actual system. This change, combined with our redefinition of equity, allowed us to eliminate the trade-off between efficiency and equity that was intrinsic to the previous formulation of the problem.
In reference to the ability of the CEPM to implement τ-fair allocations, we can say that, as far as we know, there is no DSB that already has a system in place in which students are allowed to exchange their places. Nevertheless, there is evidence that such a reform might be legally feasible. There are related situations where agents are allowed to improve their initial allocation by exchanging their rights. This evidence is abundant both in the ranges of civil servants and in the army. For instance, civil servants are allowed to exchange their placements in Spain,12 and a similar exchange can be performed in the US Army under the
so-called Enlisted Assignment Exchanges (SWAPS).13 Similar systems (i.e., agents
can exchange goods that they do not own, but they retain some rights) can be found in some socially accepted systems, such as several international student exchange programs or a recent kidney exchange14 program. Additionally, our procedure shares some features with the proposals by the Ecole Démocratique
to reform the actual system in the French-speaking area of Belgium.15
12Art. 62 in the Spanish law that governs civil servants or Law 315/1964, B.O.E 15.02.1964. This regulation can be obtained from http://www.ua.es/oia/es/legisla/funcion.htm.
13The reader is directed to http://usmilitary.about.com/od/armyassign/a/swap for further infor-mation on this matter.
14Transplant services at the Ronald Reagan UCLA Medical Center provide some information via the web page http://transplants.ucla.edu/body.cfm?id=112
15We would like to acknowledge Estelle Cantillon for pointing out these similarities. The proposals by the Ecole Démocratique can be found in French on its web page http://www.skolo.org/spip.php?article1126&lang=fr.
References
Abdulkadiro˘glu, A., Pathak, P.A., Roth, A.E., 2009. Strategy-Proofness versus Efficiency in Matching with Indifferences: Redesigning the NYC High School Match. American Economic Review 99, 1954 – 1978.
Abdulkadiro˘glu, A., Pathak, P.A., Roth, A.E., Sönmez, T., 2005. The Boston Public School Match. American Economic Review, Papers and Proceedings 95, 368 – 371.
Abdulkadiro˘glu, A., Pathak, P.A., Roth, A.E., Sönmez, T., 2006. Changing the Boston School Choice Mechanism: Strategy-Proofness as Equal Access. Mimeographed. Harvard University.
Alcalde, J., 1996. Implementation of Stable Solutions to Marriage Problems. Journal of Economic Theory 69, 240 – 254.
Alcalde, J., Romero-Medina, A., 2011a. Efficiency or Equity? A Conciliating Approach for School Choice Problems. SSRN Working Paper n. 1760592. Alcalde, J., Romero-Medina, A., 2011b. Re-reforming the Bostonian System:
A Novel Approach to the School Allocation Problem. MPRA Paper 28206, University Library of Munich, Germany.
Aumann, R.J., Maschler, M.B., 1964. The Bargaining Set of Cooperative Games, in: Dresher, M., Shapley, L.S., Tucker, A. (Eds.), Advances in Game Theory, Annals of Mathematics Study 52. Princeton University Press, Princeton, pp. 443 – 476.
Balinski, M., Sönmez, T., 1999. A Tale of Two Mechanisms: Student Placement. Journal of Economic Theory 84, 73 – 94.
Biró, P., 2008. Student Admissions in Hungary as Gale and Shapley Envisaged. TR-2008-291, Department of Computing Science, University of Glasgow. Ehlers, L., 2008. Truncation Strategies in Matching Markets. Mathematics of
Erdil, A., Ergin, H., 2008. What’s the Matter with Tie-Breaking? Improving Efficiency in School Choice. American Economic Review 98, 669 – 689. Feng, N., 2005. The Tale of Two High School Admission Mechanisms -Beijing
and Shanghai. Mimeo, University of Beijing.
Gale, D., Shapley, L.S., 1962. College Admissions and the Stability of Marriage. American Mathematical Monthly 69, 9 – 15.
Kesten, O., 2010. School Choice with Consent. Quarterly Journal of Economics 125, 1297 – 1348.
Martínez, R., Massó, J., Neme, A., Oviedo, J., 2001. On the Lattice Structure of the Set of Stable Matchings for a Many-to-One Model. Optimization 50, 439 – 457.
Roth, A.E., Rothblum, U.G., 1999. Truncation Strategies in Matching Markets: In Search of Advice for Participants. Econometrica 67, 21 – 43.
Shapley, L.S., Scarf, H., 1974. On Cores and Indivisibility. Journal of Mathemat-ical Economics 1, 23 – 37.
Shelim, T., Salem, S., 2009. Student Placement in Egyptian Colleges. MPRA Paper No. 17596.
Ting, H., 2007. Economic Analysis of Taiwan University Access System. PhD Tesis, Sun Yat-Set University.
Zhou, L., 1994. A New Bargaining Set of an N-Person Game and Endogenous Coalition Formation. Games and Economic Behavior 6, 512 – 526.
APPENDIX
A. The CEPM: An Example
This appendix provides an example illustrating how to compute the CEPM. Let us consider the following SCP. S = {1,2,3,4,5,6,7,8}; C = {a,b,c,d}; the capacity for each school is2; and the Rankings and Priorities matrices are
Φ = 2 1 3 4 2 4 1 3 3 2 1 4 4 2 3 1 1 4 2 3 1 2 3 4 1 2 4 3 2 1 3 4 ; and Π = 2 3 8 1 7 4 6 5 6 2 1 5 8 4 3 7 7 5 6 8 2 3 1 4 8 5 3 4 1 2 7 6 . The SOSM
The application of the student-proposed deferred acceptance algorithm, the output of which isµSO, is summarized in the following table16
16A row in the table indicates the applications that each school receives at each step. The students framed in a box are those whose applications are refused, whereas the remaining students are tentatively accepted by the school.
Step a b c d 1 5 ,6,7 1,8 2,3 4 2 6,7 1,8 2, 3 ,5 4 3 6,7 1,3, 8 2,5 4 4 6, 7 ,8 1,3 2,5 4 5 6,8 1 ,3,7 2,5 4 6 1,6, 8 3,7 2,5 4 7 1,6 3,7 2 ,5,8 4 8 1,2, 6 3,7 5,8 4 9 1,2 3, 6 ,7 5,8 4 10 1,2 3,7 5,6, 8 4 11 1,2 3,7 5,6 4,8 µSO := 1,2 3,7 5,6 4,8
Therefore, the SOSM proposes to allocate a placement in schoola to students1
and 2; students3and 7are accepted by school b; students5 and6are placed at schoolc; and, finally, students4and8will attend schoold.
The Compensating Placing Market
As we mention in Section IV, a Placing Market is determined by the set of stu-dents, S = {1,2,3,4,5,6,7,8}, and a preference profile, P, denoting, for each student, how she orders her “rivals” depending on an initial matching,µ. We in-terpret the preferences determined by a student asher inclination to exchangethe placement that µassigns to her with the other students. The idea of how to build these preferences is guided by the following two principles.
(a) A student participates in this market if she obtains a positive net profit from the exchange; and
(b) A student, when exchanging, wants to maximize her utility (i.e., she tries to secure a placement at the best school, according to her opinion about the educational institutions).
In the case of CEPM, the initial matching is µSO, and a student orders two
agents, other than her, who have been located at the same school by “reversing the schools ordering” proposed by the priority lists.
To illustrate how to compute the students’ preferences for exchanging,(Pci)ci∈S,
we will concentrate on student6. The process is the following
(1) Because her preferred school is a, and µSO(a) = {1,2}, we note that her two “tops for exchanging” are students 1 and 2. Moreover, because 2 =
πa1 < πa2 = 3, we note that
2P61.
(2) Now, the second school in student6’s preference list isb. Given thatµSO(b) =
{3,7}, we note that these students will be placed in the third and fourth po-sitions according toP6. Because1 =πb3 < πb7 = 3, it follows that
7P63.
(3) Because the third school in student 6’s preference list is c, and µSO(c) =
{5,6}, Principle(a)above indicates that
6P65.
For our purposes, once student6has established her own position onP6, we do not need to continue describing how the remaining students are ordered. Nevertheless, for the sake of completeness, we will also explain how P6 orders students4and8.
(4) BecauseµSO(4) =µSO(8) =d, and4 =π
d4 < πd8 = 6, we note that
Summarizing, the preferences for exchange exhibited by student6are
2P6 1P6 7P6 3P66P65P6 8P6 4, or equivalently,
P6 := 2, 1, 7, 3, 6, 5, 8, 4.
Applying a similar argument to the remaining students, we can compute P, the description of which is
P1 := 7, 3, 1, 2, 6, 5, 8, 4 P2 := 6, 5, 2, 1, 8, 4, 7, 3 P3 := 6, 5, 3, 7, 2, 1, 8, 4 P4 := 4, 8, 7, 3, 6, 5, 2, 1 P5 := 2, 1, 5, 6, 8, 4, 7, 3 P6 := 2, 1, 7, 3, 6, 5, 8, 4 P7 := 2, 1, 7, 3, 8, 4, 6, 5 P8 := 7, 3, 6, 5, 8, 4, 2, 1 Therefore,CP M µSO
= (S, P), whereS ={1, . . . ,8}is the set of students andP is the preferences profile described above.
The Tops Trading Cycle rule
To continue with our illustrative example, we now compute the outcome for the TTC rule when applied to the Compensating Placing Market that was previously described.
Step 1. At the first step, the set of students coincides with S. To draw the next digraph, we proceed as follows. A node is assigned to each student. We draw an arc connectingsitoshif the latter is the top forPsi.
17
17For instance, because7is the top forP
1, we draw an arc that departs from1and is incident to 7.
1 2 3 7 5 4 6 8
Let us observe that the above digraph contains two cycles; the first includes students2and6, and the second contains only student 4. Therefore, these three students exit from the market, and we can describe a “new market” whose students areS2 ={1,3,5,7,8}
Step 2. At the second step, the set of students coincides withS2 described above. We proceed in a similar manner to that explained in the previous step and draw its corresponding directed graph.
1 3
7
5
8
There is clearly a cycle involving students 1and 7. Therefore, K(S2) = {1,7}, and both students exit from the market, allowing us to consider the residual marketS3 ={3,5,8}.
Step 3. In this step, the set of remaining students isS3. It is easy to see that there are two cycles; one cycle contains student 3, whereas the other cycle only includes student5. Therefore,K(S3) ={3,5}andS4 ={8}. To conclude, Step 4. Because S4 is a singleton, we note that K(S4) = {8} and that S5 = ∅. Given that each student is, at this point, involved in some cycle, the algo-rithm stops.
Therefore, the outcome of the Tops Trading Cycle, applied toCP M µSO , is
si 1 2 3 4 5 6 7 8
T T C(si) 7 6 3 4 5 2 1 8
The Compensating Exchange Places Mechanism
To conclude the process, we must move from the initial matching µSO to the
matching that results from the exchange of places suggested by the Tops Trading Cycle rule. For instance, becauseT T C(1) = 7, then
µCEP(1) =µSO(7) =b.
Therefore, the outcome of the CEPM for this example is
si 1 2 3 4 5 6 7 8
µCEP(s
i) b c b d c a a d
To conclude this example and with the aim of presenting a comparison for the application of some allocation procedures relative to the data proposed in the present example, let us consider Table 1. This table assigns each student two
1 2 3 4 5 6 7 8
BM b 1 c 1 c 1 d 1 d 3 b 1 a 1 b 1
SOSM a 2 a 2 b 2 d 1 c 2 c 3 b 2 d 4
CEPM b 1 c 1 b 2 d 1 c 2 a 1 a 1 d 4
Table 1: Comparing School Choice Mechanisms
items: the school in which she secures a placement (first item) and the position of this school in the student’s ranking (second item).
Let us note that,
(1) The solution proposed by the BM is not τ-fair. In fact, the pair 5, µSO constitutes ajustified fair objectiontoµBM;
(2) When comparing theSOSMand theCEPM, it is clear that the latter Pareto dominates the former. Moreover, both mechanisms yield λ-equitable allo-cations.
B. A Proof for Proposition 4
To prove Proposition 4, let us consider the following School Allocation Problem.
S = {1,2,3};C = {a, b, c};Q = (1,1,1); and the ranking and priorities
matrices are Φ = 1 3 2 2 3 1 2 3 1 and Π = 3 2 1 2 1 3 1 3 2 .
Note that there is only one stable matching in this type of problem,µ, such that
µ(1) = c;µ(2) = b; andµ(3) = a. Nevertheless,µfails to be Pareto efficient becauseµ0, defined asµ0(1) = a;µ0(2) = b; andµ0(3) = c, Pareto dominatesµ.
C. A Proof for Theorem 17
This appendix introduces a formal proof for Theorem 17, which characterizes the set ofτ-fair allocations associated with each SCP. Let us remember that our result establishes that a matching, µ, is τ-fair if it satisfies two properties. The first property is Pareto efficiency, which is also required by Definition 10; the second is that each student (weakly) prefers the placement thatµassigns to her over the placement suggested by the SOSM,
ρsiµ(si)≤ρsiµSO(si). (1)
Proof of Theorem 17
Let(S,C,Φ,Π, Q)be a SCP, and letµbe a matching. We first show that ifµ
is τ-fair, then it satisfies Condition (1). Note that by Definition 10, µshould be Pareto efficient.
To meet our objective, let us assume by way of contradiction thatµdoes not satisfy Condition(1). There should be a student,si, such that
ρsiµSO(s
i) < ρsiµ(si).
In this case, the pair si, µSO
constitutes a justified, fair objection to µ, which contradicts our hypothesis thatµisτ-fair.
In contrast, let us assume that µis a Pareto-efficient allocation that satisfies Condition(1). We will see thatµisτ-fair.
To reach a contradiction, let us assume thatµis not τ-fair. Then, because µ
is Pareto efficient, it should be a student-matching pair,(si, µ0), that constitutes a
justified, fair objection toµ.
Therefore, by Definition 6 and Condition(1), we note that
ρsiµ0(si) < ρsiµ(si)≤ρsiµSO(si). (2)
Note that this relationship implies thatµ0(si)∈ C. Let us denote such a school
bycj.
ρsiµSO(si) for some student,si, thenµˆis not equitable. In particular, this implies
that matching µ0 fails to be equitable. Therefore, there should be a student,sh,
and a school,ck, such that
(1) ρshck < ρskµ0(sh), and
(2) πcksh < πcksl for somesl6=sh, or|µ
0(c
k)|< qck.
Therefore, if we consider any matching, µ00, such that µ00(sh) = ck, the pair
(sh, µ00)describes a counter objection toµ, which contradicts the hypothesis that
µfails to beτ-fair.
D. A Proof for Theorem 16
To prove Theorem 16, let us consider a School Allocation Problem,(S,C,Φ,Π, Q), and letµSObe its student optimal stable matching.
Let us observe that when describing the Compensating Places Market, the preferences for exchange held by a student, si, such that µ(si) 6= si, verify that
for any other studentsh,
shPsisi ⇒ρsiµ(sh) < ρsiµ(si).
By construction, the Tops Trading Cycle satisfies the requirement that for each Placing Market,(S, P), and anysi ∈Ssuch thatT T Csi(S, P)6=si,
T T Csi(S, P)Psisi.
This implies that for each student,si
ρsiµCEM(s
i) ≤ρsiµSO(si). (3)
BecauseµSO is equitable, we note the following.
i.e.,
n > X
cj∈C
qcj, and
(b) If there is some school,cj, that has vacancies atµSO, i.e.,
µSO(cj)
< qcj,
then for each student,si,
ρsiµSO(si)≤ρsicj.
To complete our proof, let us assume thatµCEM is notτ-fair. Taking into account
Equation(3)and Theorem 17, we note thatµCEM should not be Pareto efficient.
Therefore, there should be a matching,µ, such that for each student,si ∈ S
ρsiµ(si) ≤ρsiµCEM(si), and
there should be a student,sh, such that
ρshµ(sh) < ρshµCEM(sh). (4)
Let S denote the set of students fulfilling Condition (4). By Equation (3) and given thatµSO is equitable, we note that for eachs
i ∈S, there is another student,
sh, inSsuch thatµ0(si) =µCEM(sh). This finding implies thatsi’s preferences
for exchange satisfy the requirement that
shPsiT T Csi CP M µ
SO
. (5)
To reach a contradiction, for each studentsi ∈S, we lett(si)denote the stage
at which it is determinedT T Csi CP M µ
SO
. Without loss of generality, let us assume thatsi ∈S is such thatt(si)≤ t(sh)for eachsh ∈ S. By Equation(4),
we note that in the digraph for staget(si), there is no arc fromsitoµCEM(si).
This finding constitutes a contradiction, which points out that there is no 32
matching,µ, Pareto dominatingµCEM.
Ivie
Guardia Civil, 22 - Esc. 2, 1º 46020 Valencia - Spain Phone: +34 963 190 050
Fax: +34 963 190 055 Department of Economics
University of Alicante Campus San Vicente del Raspeig
03071 Alicante - Spain Phone: +34 965 903 563 Fax: +34 965 903 898 Website: www.ivie.es E-mail: [email protected]
